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Group Decision and Negotiation

Erschienen in:

06.04.2018

Coalition Formation and Stability

verfasst von: Antonio Magaña, Francesc Carreras

Erschienen in: Group Decision and Negotiation | Ausgabe 3/2018

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Abstract

This paper aims to develop, for any cooperative game, a solution notion that enjoys stability and consists of a coalition structure and an associated payoff vector derived from the Shapley value. To this end, two concepts are combined: those of strong Nash equilibrium and Aumann–Drèze coalitional value. In particular, we are interested in conditions ensuring that the grand coalition is the best preference for all players. Monotonicity, convexity, cohesiveness and other conditions are used to provide several theoretical results that we apply to numerical examples including real-world economic situations.

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A suggestion for which we are grateful to a reviewer.

A survey that shows the impact of the Shapley value in several scientific disciplines is due to Moretti and Patrone (2008).

In a previous work Hart and Kurz (1983), they defined two more notions, \(\alpha \)-stability and \(\beta \)-stability, for NTU cooperative games, but we will restrict our study to TU games.

Here \(u_T\) will not mean the unanimity game associated to coalition T.

The difference between the \(\gamma \)-model and the \(\delta \)-model lies here. In the \(\delta \)-model, Hart and Kurz assume that, if only some members of a coalition choose it, the subcoalition consisting of these members forms, while the others become singletons. Of course, if a coalition forms in the \(\gamma \)-model it also forms in the \(\delta \)-model, but the converse is not true. Here we prefer using the \(\gamma \)-model solely because the subcoalition might have a utility no longer interesting to its members.

The lack of solution appears here and in other examples below. Following Segal (2003), we could call collusion proof to any game where this occurs. In such a game, players are reduced to form, in principle, the trivial structure \(\mathcal {B}^n\), that we have called the “disagreement point”.

In Segal (2003), the expression \([u(T\cup \{i,j\})-u(T\cup \{i\})-u(T\cup \{j\})+u(T)]\) is denoted as \(\Delta _{ij}^2[u](T)\), and \(\Delta _{ij}^2\) is called the “second-order difference operator”.

This inequality is, in fact, equivalent to the convexity of the game (Ichiishi 1993, Theorem 2.1.3).

As we do not impose \(u(N)=1\), we accept as simple the null game \(u=0\), because the restriction of a simple game may well be a null subgame.

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Titel: Coalition Formation and Stability
verfasst von: Antonio Magaña
Francesc Carreras
Publikationsdatum: 06.04.2018
Verlag: Springer Netherlands
Erschienen in: Group Decision and Negotiation / Ausgabe 3/2018
Print ISSN: 0926-2644
Elektronische ISSN: 1572-9907
DOI: https://doi.org/10.1007/s10726-018-9570-1

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